The Annals of Probability

Extremal theory for long range dependent infinitely divisible processes

Gennady Samorodnitsky and Yizao Wang

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We prove limit theorems of an entirely new type for certain long memory regularly varying stationary infinitely divisible random processes. These theorems involve multiple phase transitions governed by how long the memory is. Apart from one regime, our results exhibit limits that are not among the classical extreme value distributions. Restricted to the one-dimensional case, the distributions we obtain interpolate, in the appropriate parameter range, the $\alpha$-Fréchet distribution and the skewed $\alpha$-stable distribution. In general, the limit is a new family of stationary and self-similar random sup-measures with parameters $\alpha\in(0,\infty)$ and $\beta\in(0,1)$, with representations based on intersections of independent $\beta$-stable regenerative sets. The tail of the limit random sup-measure on each interval with finite positive length is regularly varying with index $-\alpha$. The intriguing structure of these random sup-measures is due to intersections of independent $\beta$-stable regenerative sets and the fact that the number of such sets intersecting simultaneously increases to infinity as $\beta$ increases to one. The results in this paper extend substantially previous investigations where only $\alpha\in(0,2)$ and $\beta\in(0,1/2)$ have been considered.

Article information

Ann. Probab., Volume 47, Number 4 (2019), 2529-2562.

Received: March 2017
Revised: May 2018
First available in Project Euclid: 4 July 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G70: Extreme value theory; extremal processes 60F17: Functional limit theorems; invariance principles
Secondary: 60G57: Random measures

Extreme value theory random sup-measure random upper semicontinuous function stable regenerative set stationary infinitely divisible process long range dependence weak convergence


Samorodnitsky, Gennady; Wang, Yizao. Extremal theory for long range dependent infinitely divisible processes. Ann. Probab. 47 (2019), no. 4, 2529--2562. doi:10.1214/18-AOP1318.

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